Has anyone seen this video?

It's been posted here before.  I think it's a terrible idea.  Leaves produce energy.  That energy goes to the figs and the roots for storage.  Without the leaves the figs will get energy reserves from the roots and the plant will have less that it otherwise would have next Spring. 

Tried it....did not work for me

It will leave your plant less winter hardy by robbing it of sugars and starch stored in shoots and roots. Plus cuttings from those plants will be less likely to root. Also I don't see how it would cause the figs to ripen. Taking off the small figs would be a better idea and make the plant more likely to survive cold.

What about selectively removing some leaves that are shading fruit? I'm talking about maybe 1/4 of the overall? Still think it would be a bad idea?

It would sure slow down water uptake, so dehydrate ripening figs, but would this affect the rate of ripening?

Pinching the terminal bud as of 1st August helps - removing the leaves does not.
Why? Because the tree needs th leaves and will push new ones to replace the ones you removed. So you'll have new growth that will easily die-back ... in my Zone7
That's my experience at least ...

Taking the leaves off sounds like a terrible idea to me.  The leaves are what produce the sugars to ripen the figs.  Also not good to rob the tree of the leaves it needs to build its winter reserves.
To address this problem in cool climates I would do as JDSfrance said pinch the tips after the branch has 7 or 8 figs already.  Also come September I remove any figs that I don't think will ripen by the 1st hard frost day since after that the leaves will die and fall. 
IMO Figs will not ripen without leaves.  They may soften and become edible but they will not get any sweeter.

on a side note.  Grape leaf removal is a common practice but it is used to have a balanced grape plant (leaves/fruit) and mostly to provide sun light to the grape berries so they don't get mildew or rot.  They are very careful not to remove too many leaves because that would affect the sugars.

Have never done that , i will remove a few leaves directly blocking sunlight to a fig late in season if
i know it has a chance to ripen as the days are recorded via dated pictures as to how long they are
on plant.

Limiting leaf growth by pinching terminal buds is one thing, but cutting all the leaves off is a totally different situation. The tree needs its leaves to perform photosynthesis - it can't produce sugars if leaf cells can't absorb radiant energy from the sun. When the sunlight hits the leaves a series of chemical reactions take place during which the plant converts energy from the sun into energy the plant itself can use in the form of glucose - sugars - that is.

If you cut off all the leaves you'll see little buds poke out from the axes where the figs are growing or from nodes on the branches. Let the natural process take place - the fruit will ripen when it is supposed to, if you try to manipulate the process you often end up with bland or dry premature figs.

Also, when trees lose their leaves they are taking water back into the branches for the winter. That's why trees loose their leaves in times of drought and during the fall - they are trying to conserve water for when there will be little water. You don't want to cut off the natural water storage process from occurring before the tree goes dormant. I pruned all the leaves off one of my trees in the fall one year somewhat prematurely because I wanted to move it to the basement for winter storage- the tree dried out and died. I watered the tree like all my others during the winter to keep the soil moist but it still died. The plant has persisted in nature because it has a biological system that works - if the tree wasn't efficient it wouldn't survive to continue propagating. Let nature take its course.